Questions tagged [characteristic-classes]
Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.
333
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Different ways of defining the Chern character of a complex
Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form
$$
0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0,
$$
where the bundles are ...
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0
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Discrete spectrum of Dirac operator
It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so
that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete.
For example at least for $d=4$, this ...
- 2,137
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votes
2
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Even, non liftable Stiefel-Whitney class
Let $M$ be a smooth manifold and $E$ a smooth real vector bundle of even rank over $M$.
If $E$ admits of a complex vector bundle structure $\mathcal E$ ($\mathcal E_\mathbb R=E$) then all odd Stiefel-...
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Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?
The Whitney sum formula for Stiefel-Whitney classes, $w_n(V \oplus W) = \sum w_i(V) w_{n-i}(W)$, looks a lot like the one for Chern classes $c_n(V \oplus W) = \sum c_i(V)c_{n-i}(W)$. But I don't know ...
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Comparing the Segre classes of a cone with its abelian hull
Let $X$ be a smooth scheme, with a sheaf of graded quasi-coherent algebras $\mathcal{A}^*$, that yields a cone $C$ (in the sense of Fulton's intersection theory). Suppose that $\mathcal{A}^1$ is a ...
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Projectively flat connection
Let $E \to B$ be a Hermitian vector bundle. If $E$ has a projectively flat connection, then its total Chern character has the form $\mbox{ch}(E) = \mbox{rank} \cdot \exp(\mbox{slope})$. Is the ...
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Whitney sum via Gysin
Let $E_1\to E\to E_2$ be a short exact sequence of vector bundles. The Whitney sum formula says that $e(E)=e(E_1)e(E_2)$, i.e. that the Euler class is multiplicative.
Is there a proof of this fact ...
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Motivation for the definition of complex orientable cohomology theory
PRELIMINARY DEFINITIONS:
Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have:
$$
\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)
$$
So there is a special ...
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answer
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Curvature as infinitesimal holonomy 2
This question may be seen as a follow up of this original question. I'm learning Cheeger-Simons differential characters (reading Differential Characters of Bär and Becker).
If I understand correctly, ...
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de Rham-invariants of a Riemannian metric
$\DeclareMathOperator{\Sym}{Sym}$For $N>0$, consider the $O_N$-representations $V = \mathbb R^N$ and $M_n = \ker (\Sym^n{V}\otimes\Sym^2 V\to \Sym^{n+1} V\otimes V)$ (the irreducible $GL_n$-...
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answer
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Shulman's Thesis on Characteristic Classes
I am trying to find a copy of H. Shulman's 1972 Berkeley thesis 'On Characteristic Classes'. I've seen it referenced in Bott's 'On the de Rham theory of Certain Classifying Spaces' but I can't seem to ...
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The maximum number of vertical independent vector fields on the tangent bundle
Let $M$ be a differentiable manifold.
Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a name for ...
- 312
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0
answers
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Push forward of Chern character and index theorem
I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998).
I expose here the setup for my ...
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How to calculate the total chern classes of CP^n [closed]
When calculating the total chern class of $\ CP^n$, we use the fact that their is a exact sequence of vector bundles over $\ CP^n$:
$$\ 0\to S \to C^{n+1} \to Q \to 0$$
And identify the bundle $\ TCP^...
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Visualize how the 5d Dold manifold and Wu manifold are cobordant via a 6d manifolds with boundaries
Are there simple intuitions and arguments to visualize why the following two 5-manifolds are cobordant to each other with the oriented structures? (They can be two boundaries of 6-dimensional oriented ...
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The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$
$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...
- 2,137
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votes
0
answers
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Evaluating the Euler class of a circle bundle on fibers
I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle.
This might be completely obvious, but I don't see how to answer the ...
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Characteristic classes of quotient manifold
Let $M$ be a compact oriented smooth manifold with boundary and let $G$ be a compact Lie group acting smoothly, orientation-preservingly and freely on $M$.
(Under what conditions) is there a ...
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Differential refinement of homology
Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism ...
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votes
1
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Calculating topological $K(X)$ for complex projective manifolds
In the introduction to the book Vector bundles and K-theory
http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html
two approaches to classification of (topological) vector bundles are discussed - the ...
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Chern classes of a mapping torus vector bundle in terms of the construction data
Let $\pi:E\to X$ be a complex vector bundle*, and $f:E\to E$ a bundle isomorphism.
Consider the mapping torus
$$E(f) := \frac{E\times [0,1]}{E \times \{0\}\sim_f E \times \{1\}}$$
where the ...
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Relation between Bott-Chern forms and Second fundamental form
Given a short exact sequence of holomorphic Hermitian vector bundles
$$0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0,$$
the second fundamental form measures the obstruction of $E\simeq F\oplus ...
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answers
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Which classes in $\mathrm{H}^4(B\mathrm{Exceptional}; \mathbb{Z})$ are classical characteristic classes?
Let $G$ be a compact connected Lie group. Recall that $\mathrm{H}^4(\mathrm{B}G;\mathbb{Z})$ is then a free abelian group of finite rank. Let us say that a class $c \in \mathrm{H}^4(\mathrm{B}G;\...
- 53.1k
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votes
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Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$
For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e.,
$$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
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Instanton numbers for diverse gauge bundles on diverse manifolds --- their relations to characteristic classes
It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$
$$
n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...
- 10.3k
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votes
1
answer
510
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Realizing Stiefel-Whitney classes via vector bundles
Let $X$ be a CW complex. If $E$ is a vector bundle over $X$, then it's well-known that the Stiefel-Whitney classes $w_j(E) \in H^j(X,\mathbb F_2)$ of $E$ are determined from the classes $w_{2^k}(E)$ (...
- 61.5k
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Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?
This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case.
Let $M$ be a ...
- 1,596
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Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
- 2,169
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votes
0
answers
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Equivalence between integrals over a reduced space
Context: I have been trying to understand this paper from Y. Cho and K. Kim. More precisely, a specific argument in Lemma 2.2 where they say the ABBV localization formula on an integral over a ...
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0
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Exponential of mixed-type End-valued differential form
Let $E\rightarrow \mathbb{P}^1$ be a complex vector bundle and let $a_{(0,0)},a_{(1,0)},a_{(0,1)},a_{(1,1)}$ be differential forms
such that $a_{(i,j)}\in\Omega^{i,j}(\mathbb{P}^1,End(E))$. I would ...
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votes
1
answer
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Are all classes Stiefel-Whitney classes?
When I thought of this question, I was sure it must have been asked before on this site, but I could't find anything. Maybe my search skills are lacking, or maybe the question is obvious and it's my ...
- 1,131
6
votes
1
answer
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Action of Steenrod algebra on Chern classes
This is question about result from Brown and Peterson $H^*(MO)$ as an algebra over the Steenrod algebra. Unfortunately, the paper is not available on the Internet, so I can't find the proof.
One of ...
user131113
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4
answers
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Analogy between Stiefel-Whitney and Chern classes
There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field $\mathbb R$ with $\mathbb C$, coefficient ring $\mathbb Z/2$ with $\mathbb Z$ and scales the grading ...
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votes
1
answer
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Derivative of the Bott-Chern forms
The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
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votes
1
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Definition of 1st degree obstruction class
Recently I go through obstruction class illustrated by Milnor.
He defined $\mathfrak{o}_i$by an element in $H^i(M; \pi_{i-1}(V_{n-i+1}(F))$, which is cohomology with local coefficients.
But the 0th ...
- 1,084
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If $n$ is not a power of 2 then the dual Stiefel-Whitney class $\bar{w}_{n-1} = 0$
Stiefel-Whitney classes are invertible and for $w$, the Stiefel-Whitney class of the tangent bundle of $M$, we have its inverse $\bar{w}$. I want to prove that if $n$ is not a power of 2 then the dual ...
- 1,084
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Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$
An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below:
Characteristic classes are certain cohomology classes associated
...
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2
votes
1
answer
493
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Relation between compact vertical cohomology and local cohomology groups
I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:
The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, ...
5
votes
0
answers
251
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Integration on an non-orientable manifold [closed]
Suppose $M_n$ is a $n$ dimensional non-orientable manifold.
I am interesting in knowing whether the following statements are true:
A characteristic class $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$...
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votes
1
answer
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Bordism groups of $X$, Thom isomorphism and characteristic numbers
Recap: bordism group
An oriented singular $n$-manifold in $X$is a map $f:M^n\to X$ where $M$ is a finite disjoint union of $n$-dimensional smooth manifolds.
The empty set is an admissible oriented ...
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votes
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Can one relate $g(\nabla_XX,\nabla_YY) = |\nabla_XY|^2$ with some characteristic class?
Let $(M^2,g)$ be a closed surface and let $X,Y\in C^{\infty}(TM)$ such that $[X,Y] = 0$. I am working on a problem where I have to deal with the following term
$$g(\nabla_XX,\nabla_YY) - |\nabla_XY|^...
- 1,774
3
votes
1
answer
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Chern classes of complex vector bundle
I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:
$E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...
4
votes
1
answer
267
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Chern -Weil map for topological principal G bundles
Let $G$ be a Lie group.
In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following :
The notion of a topological principal $G$...
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11
votes
3
answers
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A binary operation on vector bundles that adds Chern classes?
Let $E$ and $F$ be two complex vector bundles over a space $X$. There's a fairly well-known binary operation called the direct sum, written $E\oplus F$, which has the property that its first Chern ...
- 1,131
17
votes
0
answers
474
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Can the intermediate Chern classes be expressed as Euler classes?
General question: We know that the top Chern class $c_n(\xi)$ of an $n$-dimensional complex vector bundle $\xi$ is its Euler class, while the first Chern class, $c_1(\xi)$, is the Euler class of its ...
- 2,052
2
votes
0
answers
454
views
Splitting principle for real vector bundles
I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck at Lemma 2.27.
i) How does this lemma show that a real vector bundle can be given by a pullback of ...
- 145
3
votes
0
answers
128
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About the proof of Milnor-Novikov theorem about multiplicative generators of (complex) bordism ring
I am trying to understand part of Milnor-Novikov theorem about multiplicative generators of $MU_* \cong \mathbb Z[x_1, x_2, \dots]$ using S.Kochman’s “Bordsim, Stable Homotopy and Adams Spectral ...
user131113
6
votes
3
answers
606
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Electromagnetism as a $U(1)$-gauge theory
I would like to learn gauge theory, starting from the simplest case. I have heard that I should start with electromagnetism, which is just the $U(1)$-gauge theory. All the references I know are ...
- 5,038
2
votes
1
answer
208
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Pontryagin square of first Stiefel-Whitney class
Let $w_1$ be the 1st Sitefel-Whitney classes of the tangent bundle of a 4-manifold $M$ ($M$ is non orientable). My question is
Is $$\exp\left(\frac{i\pi}{2}\int_{M_4} \mathcal{P}(w_1^2)\right)$$ ...
- 487
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votes
0
answers
169
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Understanding $w_2$ as an obstruction to trivializing the tangent bundle over 2-cells
I am reading through "A Geometric Proof of Rochlin's Theorem", and it is occurring to me, again, that I don't understand spin structures / $w_2$. My confusion arrises in, naturally, the proof of ...
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